Why is the denominator 2bc in the cosine rule?

The two sides adjacent to angle A are b and c, and the formula comes from the dot product, so the denominator is always 2bc.

What is the fastest route from three sides to the area?

Heron: s = (a + b + c) / 2 and area = sqrt(s(s-a)(s-b)(s-c)), with no angle needed.

What is the ambiguous case?

With two sides and a non-included angle, if the side opposite the given angle is shorter than the other side but longer than the altitude b sinA, two angles are possible.

Why does the parallelogram diagonal use a plus sign?

The diagonal faces the supplement of the included angle, and the cosine changes sign, turning the minus into a plus.

S4 Maths sine and cosine rule paper 2: reverse work and applications

2026-11-10

Tutor Prep Guide · S4 Maths Sine and Cosine Rule

Ten items on reverse angles, area from three sides and parallelogram diagonals, including the two-solution case.

Direct answers　Items 01 to 03 reverse the two rules, 04 to 06 area, Heron and a parallelogram, 07 to 09 the ambiguous case, finding the third angle first and the 13-14-15 area, and 10 a wrong sign in the cosine rule.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1
Chapter	Chapter: S4 Maths Sine and Cosine Rule
Level	S4

02　Questions (10 items)

FoundationIn a triangle a = 10 cm, A = 45 and B = 60. Find b.
FoundationIn a triangle b = 9 cm, c = 5 cm and A = 120. Find a.
FoundationThe three sides of a triangle are a = 7 cm, b = 8 cm and c = 9 cm. Find A.
IntermediateTwo sides of a triangle are 8 cm and 10 cm and the area is 20 cm². Find all possible values of the included angle C.
IntermediateThe sides of a triangle are 5 cm, 6 cm and 7 cm. Find its area.
IntermediateA parallelogram has sides 6 cm and 8 cm with an angle of 60 degrees between them. Find its longer diagonal.
ChallengeIn a triangle a = 10 cm, b = 12 cm and A = 35. How many solutions are there, and what is B?
ChallengeIn a triangle A = 30, B = 105 and c = 8 cm. Find a.
ChallengeThe sides of a triangle are 13 cm, 14 cm and 15 cm. Find its area.
Error redoA student writes a^2 = b^2 + c^2 + 2bc cosA and gets a = 10.4 cm for b = 5, c = 7 and A = 60. What is wrong, and what is the correct value?

03　Answers and one-line notes

#	Answer	One-line note
1	12.2 cm	b = a sinB / sinA = 10 x sin60 / sin45 = 12.2 cm; each side must be paired with its own opposite angle.
2	12.3 cm	a^2 = 81 + 25 – 2 x 45 x cos120 = 106 + 45 = 151, so a = 12.3 cm; the obtuse angle makes the subtracted term positive.
3	48.2 degrees	cosA = (64 + 81 – 49) / (2 x 8 x 9) = 0.667, so A = 48.2 degrees; the denominator is 2bc, never 2a.
4	30 or 150 degrees	sinC = 2 x 20 / (8 x 10) = 0.5, so C is 30 or 150 degrees and both give the same area.
5	14.7 cm²	s = (5 + 6 + 7) / 2 = 9, so the area is sqrt(9 x 4 x 3 x 2) = 14.7 cm².
6	12.2 cm	d^2 = 36 + 64 + 2 x 48 x cos60 = 148, so d = 12.2 cm; the longer diagonal of a parallelogram uses a plus sign.
7	Two solutions: B = 43.5 or 136.5 degrees	The altitude h = b sinA = 6.9 cm is less than a = 10 cm while a is less than b, so there are two solutions, B = 43.5 or 136.5 degrees.
8	5.66 cm	C = 180 – 30 – 105 = 45 degrees, then a = c sinA / sinC = 8 x sin30 / sin45 = 5.66 cm; find the third angle first.
9	84 cm²	s = 21, so the area is sqrt(21 x 8 x 7 x 6) = 84 cm²; the Heron route is faster than finding an angle first.
10	The middle sign must be minus; a = 6.24 cm	a^2 = 25 + 49 – 35 = 39, so a = 6.24 cm; the plus sign belongs only to the diagonal of a parallelogram.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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